Super-orthogonal space-time trellis codes, and applications thereof

ABSTRACT

A super set of orthogonal space-time block codes is combined with set partitioning to form super-orthogonal space-time trellis codes having full diversity, enhanced coding gains, and improved rates. In communications systems, these codes are implemented by an encoder of a diverse transmitter to send an information signal to a receiver having one or more receiver elements. A decoder in the receiver decodes the encoded signal to reproduce the information signal. A method of the invention is used to generate set portioning structures and trellis structures that enable code designers to systematically design the codes of the invention.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.09/985,886, filed Nov. 6, 2001, which claims the benefit of U.S.Provisional Application No. 60/312,781, filed Aug. 17, 2001, and U.S.Provisional Application No. 60/246,426, filed Nov. 6, 2000, each ofwhich is incorporated herein by reference in its entirety.

FIELD OF THE INVENTION

The present invention relates to telecommunications. More particularly,it relates to wireless telecommunications systems.

BACKGROUND OF THE INVENTION

It is well known that using a diversity scheme can improve thesignal-to-noise ratio of a received information signal in atelecommunications system. A diversity scheme involves, for example,using information derived from several signals transmitted overindependent fading paths to produce a single received informationsignal. In telecommunications systems having multiple transmit antennas(transmit diversity), a coding scheme can be used to encode thetransmitted information and thereby improve reliability. Using a codingscheme with such telecommunications systems reduces errors in thereceived information signal.

The use of codes for improving the reliability of a telecommunicationssystems having multiple transmit antennas has been discussed inarticles. For example, a code for providing full diversity fortelecommunications system using two transmit antennas is discussed in anarticle by Alamouti. See S. M. Alamouti, “A simple transmitter diversityscheme for wireless communications,” in IEEE Journal on Selected Areasof Communications, Vol. 16, pp. 1451-1458, November 1998, which isincorporated herein by reference in its entirety. The code of Alamoutiis generalized to any number of antennas in an article by Tarokh et al.See V. Tarokh et al., “Space-time block codes from orthogonal designs,”in IEEE Trans. Inform. Theory, Vol. 45, pp. 1456-1467, July 1999, whichis incorporated herein by reference in its entirety. As suggested by thetitle, the code of Tarokh et al. is referred to as a space-time blockcode.

In their article, Tarokh et al. have generalized the theory oforthogonal designs to show when a full diversity scheme is possible andhow to achieve it. See id. Although the space-time block code of Tarokhet al. provides full diversity, its main goal is not to provide enhancedcoding gains. See id., and V. Tarokh et al., “Space-time block codes forwireless communications: performance results,” in IEEE Journal onSelected Areas of Communications, Vol. 17, pp. 451-460, March 1999,which is incorporated herein by reference in its entirety. To achieveenhanced coding gains, one needs to concatenate another code to thespace-time block code of Tarokh et al.

Space-time trellis codes combine a trellis code with an inner space-timeblock code to provide enhanced coding gain. Space-time trellis codes arediscussed in an article by Alamouti et al. See S. Alamouti et al.,“Trellis-coded modulation and transmit diversity: design criteria andperformance evaluation,” in IEEE International Conference on UniversalPersonal Communications (ICUPC-98), Vol. 2, pp. 917-920, 1998, which isincorporated herein by reference in its entirety. The space-time trelliscoding scheme of S. Alamouti et al. is used for Rayleigh fading channelswith large space-time correlations in an article by Siwamogsatham et al.See S. Siwamogsatham et al., “Robust space-time coding for correlatedRayleigh fading channels,” Allerton Conference, October 2000, which isincorporated herein by reference in its entirety.

As will be understood by a person skilled in the relevanttelecommunications art, the coding scheme of Alamouti et al. andSiwamogsatham et al. does not produce the maximum possible coding gain,nor does it provide a method for designing new codes for systems otherthan those discussed. For example, it is not clear how to design codesfor telecommunications systems having different number of states ordifferent rates than the ones discussed by Alamouti et al. andSiwamogsatham et al. Thus, there is a need for new codes having improvedcoding gains and a method for designing such codes.

BRIEF SUMMARY OF THE INVENTION

A super set of orthogonal space-time block codes is combined with setpartitioning to form super-orthogonal space-time trellis codes havingfull diversity, enhanced coding gains, and improved rates. Incommunications systems, these codes are implemented by an encoder of adiverse transmitter to send an information signal to a receiver havingone or more receiver elements (e.g., one or more antennas or opticalreceivers). A decoder in the receiver decodes the encoded signal toreproduce the information signal. The codes of the invention can beimplemented by any receiver and any transmitter having at least twotransmission elements (e.g., antennas or optical receivers).

In embodiments, the invention is used to encode and decode informationsignals sent between two communication devices. For example, inembodiments, the invention is implemented by cell phone communicationssystems, TV and video communications systems, and digital area networkcommunications systems. The invention can be used, for example, forcommunications systems transmitting in the radio and/or opticalfrequency ranges.

A method of the invention is used to generate set portioning structuresand trellis structures that enable code designers to systematicallydesign the codes of the invention.

Further features and advantages of the present invention, as well as thestructure and operation of various embodiments of the present invention,are described in detail below with reference to the accompanyingdrawings.

BRIEF DESCRIPTION OF THE DRAWINGS/FIGURES

The present invention is described with reference to the accompanyingfigures. In the figures, like reference numbers indicate identical orfunctionally similar elements. Additionally, the leftmost digit ordigits of a reference number identify the figure in which the referencenumber first appears. The accompanying figures, which are incorporatedherein and form part of the specification, illustrate the presentinvention and, together with the description, further serve to explainthe principles of the invention and to enable a person skilled in therelevant art to make and use the invention.

FIG. 1 illustrates an telecommunications system according to anembodiment of the invention.

FIG. 2 illustrates a transmitter according to an embodiment of theinvention.

FIG. 3 illustrates a receiver according to an embodiment of theinvention.

FIG. 4 illustrates a digital networking system according to anembodiment of the invention.

FIG. 5A illustrates an encoder according to an embodiment of theinvention.

FIG. 5B illustrates a second example encoder according to an embodimentof the invention.

FIG. 6 illustrates a four-state trellis according to an embodiment ofthe invention.

FIG. 7 illustrates a set partitioning for a BPSK telecommunicationssystem according to an embodiment of the invention.

FIG. 8 illustrates a set partitioning for a QPSK telecommunicationssystem according to an embodiment of the invention.

FIG. 9 illustrates a set partitioning for an 8-PSK telecommunicationssystem according to an embodiment of the invention.

FIG. 10 illustrates a four-state trellis for designing a BPSK or a QPSKtelecommunications system according to an embodiment of the invention.

FIG. 11 illustrates a four-state trellis for designing an 8-PSKtelecommunications system according to an embodiment of the invention.

FIG. 12 illustrates a two-state trellis for designing a BPSK or a QPSKtelecommunications system according to an embodiment of the invention.

FIG. 13 illustrates an eight-state trellis for designing an 8-PSKtelecommunications system according to an embodiment of the invention.

FIG. 14 illustrates a four-state trellis for designing an 8-PSKtelecommunications system according to an embodiment of the invention.

FIG. 15 illustrates simulation results for BPSK telecommunicationssystem according to embodiments of the invention.

FIG. 16 illustrates simulation results for QPSK telecommunicationssystem according to embodiments of the invention.

FIGS. 17A-B illustrate a flowchart of the method of the invention fordesigning super-orthogonal space-time trellis codes.

DETAILED DESCRIPTION OF THE INVENTION

As described herein, the present invention provides new structures andmethods for designing space-time trellis codes having full diversity andenhanced coding gain. Theses enhanced codes are referred to assuper-orthogonal space-time trellis codes. As will be understood by aperson skilled in the relative communications art, the enhanced codes ofthe invention are use in telecommunication systems that includetransmitters having multiple transmission elements (e.g., at least twoantennas or optical transmitters) and receivers having one or morereceiver elements (e.g., at least one antenna or optical receiver).

System Embodiments of the Invention

FIG. 1 illustrates an example telecommunications system 100 thatembodiments the invention. Telecommunication system 100 includes atransmitter 104 and two receivers 112 and 118.

Transmitter 100 has at least two transmission elements or antennas. Inan embodiment, transmitter 104 has exactly two antennas 108 and 110.Transmitter 104 receives as an input an information signal 102.Information signal 102 is encoded by transmitter 104 using a decoder106. The operation of decoder 106 is described in detail below. Asdescribed more fully below, transmitter 104 transmits a first encodedsignal from antenna 108 and a second encoded signal from antenna 110 attime T₁ Transmitter 104 then transmits a third encoded signal fromantenna 108 and a fourth encoded signal from antenna 110 at time T₂.

Receiver 112 receives all four encoded signal transmitted by transmitter104 using an antenna 116. These signal are decoded using a decoder 114.The output or receiver 112 is a reproduced information signal 102. Theoperation of receiver 112 is further described below with reference toFIG. 3.

Receiver 118 has at least two receiver elements or antennas. In anembodiment, receiver 118 has exactly two antennas 122 and 124. The firstand third encoded signals are received using antenna 122. The second andfourth encoded signals are received using antenna 124. These fourreceived signals are decoded by a decoder 120. The output of receiver118 is also a reproduced information signal 102. As will be understoodby a person skilled in the relevant communications art, the operation ofreceiver 118 is similar to the operation of receiver 112.

FIG. 2 illustrates a example transmitter 200 according to the invention.Transmitter 200 operates in a manner similar to transmitter 104. in anembodiment, transmitter 200 has a signal input module 202, and encoder210, two pulse shapes 212 and 220, two modulation modules 214 and 222, acarrier signal module 218, and two antennas 216 and 224.

In an embodiment, signal input module 202 comprises a microphone 204 forconverting a voice signal to an analog electrical. The output signal ofmicrophone 204 is converted to a digital electrical signal usinganalog-to-digital converter (ADC) 206. The output of ADC 206 is thensupplied to an optional signal processor 208. In an embodiment, signalprocessor 206 is used to compress the digital signal output by ADC 206.

As will be understood by a person skilled in the relevant art, inputsignal module 202 is not limited to the features shown in FIG. 2. Inother embodiments of the invention, input signal module 202 comprises,for example, the digital output of a computer or other similar device.In fact, input signal module 202 can comprise any combination of devicesthat result in the generation of a stream of digital bits or data.

The output of input signal module 202 is provided to an encoder 210.Encoder 210 encodes data using a super-orthogonal space-time trelliscode, as taught herein. The operation of encoder 210 is described indetail below with reference to FIGS. 5-16.

As shown in FIG. 2, in an embodiment, encoder 210 produces two outputs.A first output is operated on by a pulse shaper module 212 and thenup-converted by modulator module 214. This output is transmitter by anantenna 216. A second output is operated on by a pulse shaper module 220and then up-converted by modulator module 222. This output istransmitter by an antenna 224. Both modulator 214 and modulator 222 aresupplied a carrier signal for a carrier module 218. As can be seen inFIG. 2, antenna 216 transmits a signal 217 and antenna 224 transmits ansignal 225 at time T₁.

The operation of the various devices of transmitter 200 will beunderstood by a person skilled in the relevant art given the descriptionof the invention contained herein.

FIG. 3 illustrates an example receiver 300 according to an embodiment ofthe invention. Receiver 300 uses a signal antenna 304 to receive asignal 302. In addition to antenna 304, receiver 300 includes adown-converter module 306, an ADC module 310, a demodulator module 312,a decoder 314, and a decoded signal processor module 316.

Down-converter module 306 down-converts the signal 302 received byantenna 304. A carrier signal is supplied to down-converter module 306by a carrier module 308. The analog output signal of down-convertermodule 306 is converted to a digital signal by ADC 310.

Demodulator 312 demodulates the received signal 302 after it isdown-converted. The demodulated signal is then supplied to decoder 314for decoding according to the invention. Decoder 314 is designed todecode the super-orthogonal space-time trellis codes described herein.

The output of decoder 314 is provided to decoded signal processor module316. As shown in FIG. 3, in an embodiment module 316 includes anoptional signal processor 318, a digital-to-analog converter (DAC) 320and a speaker 322. The output of speaker 322 is an audio signal 324.Signal processor 318 can be any desired processor such as, for example,an equalizer, a dynamic range expander, et cetera.

As will be understood by a person skilled in the relevant art, decodedsignal processor module 316 is not limited to the features shown in FIG.3. In other embodiments of the invention, decoded signal processormodule 316 comprises, for example, a computer or other similar device.In fact, decoded signal processor module 316 can comprise anycombination of devices that operate on a stream of digital bits or data.

As will be understood by persons skilled in the relevant arts, thepresent invention is not limited to systems operating at radiofrequencies. For example, as illustrated in FIG. 4, the invention can beused in systems communicating at optical frequencies. FIG. 4 shows theinvention being used to allow two digital network devices 402 and 420 tocommunicate as part of a wireless local area network 400.

In an embodiment, wireless local area network 400 comprises at least twodigital network device 402 and 420. In an embodiment, device 402 is acomputer and device 420 is a network port device. In another embodiment,both device 402 and 420 are computers or other similar digital devices.Device 402 is coupled to a transmitter 404 having an encoder 406 and twooptical transmitter elements 408 and 410. Device 420 is coupled to areceiver 412 having a decoder 418 and two optical receiver elements 414and 416. Encoder 406 encodes data received from digital network device402 using a super-orthogonal space-time trellis code. Decoder 418decodes the signals transmitted by transmitter 404 and provides adecoder signal to digital network device 420. In an embodiment, opticaltransmitter elements 408 and 410 transmit information at an infraredfrequency, and optical receiver elements 414 and 4160 receiveinformation at an infrared frequency. The operation of each of thecomponents of wireless local area network 400 will be known to a personskilled in the relevant art given the description of the inventionherein.

FIG. 5A. illustrates an example encoder 500A according to the invention.Encoder 500A is intended for use with a transmitter having “N”transmission elements. In an embodiment, encoder 500A comprises a bufferor shift register 504A, a lookup table 506A, and a transmission module512A. As described herein, encoder 500A encodes input data bits usingthe enhanced codes of the invention.

During operation of encoder 500A, a group of input data bits is shiftedinto shift register 504A. These data bits are then provided to lookuptable 506A as an input. Lookup table 506A uses the input data bits todetermine at least N variables. These variables are referred to astransmission variables. These transmission variables are supplied totransmission module 512A. Transmission module 512A implements an N×Ntransmission matrix. In an embodiment, transmission module 512 operateson the transmission variables to form signals {C₁₁, C₁₂, . . . C_(1N),C₂₁, C₂₂, . . . C_(2N), . . . C_(N1), C_(N2), . . . C_(NN)}.Transmission module 512A outputs signal C₁₁ to transmission element A₁,signal C₁₂ to transmission element A₂, and signal C_(1N) to transmissionelement A_(N) at a time T₁. Transmission module 512A outputs signal C₂₁to transmission element A_(l), signal C₂₂ to transmission element A₂,and signal C_(2N) to transmission element A_(N) at a time T₂.Transmission module 512A outputs signal C_(N1) to transmission elementA₁, signal C_(N2) to transmission element A₂, and signal C_(NN) totransmission element A_(N) at a time T_(N).

FIG. 5B. illustrates a second example encoder 500B according to theinvention. In an embodiment, encoder 500B comprises a buffer or shiftregister 504B, a lookup table 506B, and a transmission module 512B. Asdescribed below, encoder 500B encodes input data bits using asuper-orthogonal space-time trellis code. As can be seen in FIG. 5B,encoder 500B is similar in operation to encoder 500A.

During operation of encoder 500B, two or more data bits from input databit stream 502 are temporarily stored in shift register 504 at a timeT₀. These bits are divided into two groups. The first group is referredto as state bit(s) B_(A). The second group is referred to as modulationbit(s) B_(B). The location of these bits in shift register 504 isunimportant. Thus, any group of bit(s) in shift register 504 can be usedas the state bit(s) B_(A). Similarly, any group of bit(s) in shiftregister 504 can be used as the modulation bit(s) B_(A).

In an embodiment, lookup table 506 contains both modulation symbols (X1,X2) and code selection parameters (Θ). Both the state bit(s) and themodulation bit(s) stored in shift register 504 are provided to lookuptable 506 as inputs. As describe in more detail below, in an embodiment,the modulation bit(s) are used to select modulation symbols X1 and X2,and the state bit(s) are used to select the code selection parameter Θ.These symbols (X1, X2, Θ) are referred to herein as transmissionvariables.

As shown in FIG. 5B, the transmission variables output from lookup table506B are provided to transmission module 512B. Transmission module 512Boperates on the transmission variables to form four signals X₁e^(jΘ),X₂, −X₂*e^(jΘ), and X₁*. Transmission module 512B outputs signalX₁e^(jΘ) to a first transmission element A₁ and signal X₂ to a secondtransmission element A₂ at a time T₁. Transmission module 512B thenoutputs signal −X₂*e^(jΘ) to the first transmission element A₁ andsignal X₁* to the second transmission element A₂ at a time T₂.

As will be understood by a person skilled in the relevant art given thedescription herein, any of the super-orthogonal space-time trellis codesdescribed herein can be implemented using encoder 500A or encoder 500B.

Super-Orthogonal Space-Time Trellis Code Structures

In this section, the structures used to design the enhanced codes of theinvention are described. These structures are illustrated in FIGS. 7-14.

Before describing the structures used to design the codes of theinvention, it is important to clarify the notation used herein. FIG. 6illustrates an example four-state trellis 600 used to design codes. Thefour states of trellis 600 are state 1, state 2, state 3, and state 4.As described in more detail below, state 1 is associates with setpartitions {A₀₀, A₀₁}, state 2 is associates with set partitions {A₁₀,A₁₁}, state 3 is associates with set partitions {A₀₁, A₀₀}, and state 4is associates with set partitions {A₁₁, A₁₀}.

Each state of trellis 600 is represented by two nodes. State 1 isrepresented by nodes 632 and 634. State 2 is represented by nodes 636and 638. State 3 is represented by nodes 640 and 642. State 4 isrepresented by nodes 644 and 648. Transitions between the four states oftrellis 600 ca occur only along the lines that connect these nodes. Forexample, from state 1, it is possible to transition to state 1 alongline 602 or to state 2 along line 604. It is not possible, however, totransition from state 1 to state 3 or state 4 because there is no loneconnecting node 632 to node 642 or node 648.

As will be understood by a person skilled in the relevant communicationsart given the discussion herein, one or more bits are typically used todetermine transitions between the states of a trellis. For example,starting at state 1 of trellis 600, if a bit equal to 0 is received, thecurrent state will change to state 1 along line 602, and if a bit equalto 1 is received, the current state will change to state 2 along line604. Starting at state 2 of trellis 600, if a bit equal to 0 isreceived, the current state will change to state 3 along line 610, andif a bit equal to 1 is received, the current state will change to state4 along line 612. Starting at state 3 of trellis 600, if a state bitequal to 0 is received, the current state will change to state 2 alongline 608, and if a state bit equal to 1 is received, the current statewill change to state 1 along line 606. Finally, starting at state 4 oftrellis 600, if a bit equal to 0 is received, the current state willchange to state 4 along line 616, and if a state bit equal to 1 isreceived, the current state will change to state 3 along line 614. Theseare the only state changes available for trellis 600.

A second important structure used in code designs is a set partitioningstructure. A set partitioning structure identifies all possible sets ofmodulation symbols associated with a given trellis state or statetransition. For example, FIG. 7 illustrates a set partitioning structure700 according to an embodiment the invention.

A first structure used to design the enhanced codes of the invention isset partitioning structure 700. Set partitioning structure 700 isintended for use in designing codes for BPSK communication systems. Asshown in FIG. 7, there are six possible partitions of the four groups ofsymbols assigned to structure 700. Partition A₀ includes the symbols{00, 11}. These symbols are identified by tracing down the branches thatoriginate from the node labeled A₀. As described below, group {00} isone possible assignment for the transmission variables X1 and X2 (i.e,X1=0 and X2=0). Partition A₁ includes the symbols {01, 10}. Thesesymbols are also identified by tracing down the branches that originatefrom the node labeled A₁. As described above, group {01} is one possibleassignment for the transmission variables X1 and X2 (i.e, X1=0 andX2=1). The other possible partitions of structure 700 are identified ina similar manner. Partition A₀₀ includes the symbols {00}. Partition A₀₁includes the symbols {11}. Partition A₁₀ includes the symbols {01}.Partition A₁₁ includes the symbols {10}. How to use structure 700 todesign enhanced codes according to the invention is more fully describedbelow.

The assignment of states to a trellis structure is an important aspectof any code design that uses a trellis structure. Similarly, thepartitioning of symbols is an important aspect of any code design thatuses partitioning. Thus, as explained in more detail below, the statesassigned to the partitions identified by the structures in FIGS. 7-9 andto the trellis structures in FIGS. 10-14 are important aspects of theinvention. As described herein, when used in combination, thesestructures enable a person skilled in the relevant communications art todesign super-orthogonal space-time trellis codes according to theinvention.

The structures shown in FIGS. 8-14 will now be described.

FIG. 8 illustrates a set partitioning structure 800 that is intended foruse in designing codes for QPSK communication systems according to theinvention. As shown in FIG. 8, there are at least fourteen possiblepartitions of the groups of symbols assigned to structure 800. Forexample, partition A₀ includes the symbols {00, 22, 02, 20, 11, 33, 13,31}. These symbols are identified by tracing down the branches thatoriginate from the node labeled A₀. Partition A₁ includes the symbols{01, 23, 03, 21, 10, 32, 12, 30}. These symbols are also identified bytracing down the branches that originate from the node labeled A₁. Otherpossible partitions of structure 800 are identified, for example, by thelabels A₀₀, A₀₁, A₁₀, and A₁₁. How to use structure 800 to design codesaccording to the invention is described more fully below.

FIG. 9 illustrates a set partitioning structure 900 that is intended foruse in designing codes for 8-PSK communication systems according to theinvention. As shown in FIG. 9, there are at least thirty possiblepartitions of the groups of symbols assigned to structure 900. PartitionA₀₀ includes the symbols {00, 44, 04, 40, 22, 66, 26, 62, 02, 46, 06,42, 24, 60, 20, 64}. These symbols are identified by tracing down thebranches that originate from the node labeled A₀₀. Other similarpartitions include partitions A₀₁, A₁₀, and A₁₁. Partition A₀₁ includesthe symbols {11, 55, 15, 51, 33, 77, 37, 73, 13, 57, 17, 53, 31, 75, 35,71}. Partition A₁₀ includes the symbols {01, 45, 05, 41, 23, 67, 27, 63,03, 47, 07, 43, 21, 65, 25, 61}. Partition A₁₁ includes the symbols {10,54, 14, 50, 32, 76, 36, 72, 12, 56, 16, 52, 30, 74, 34, 70}. Otherpossible partitions are identified by labels in FIG. 9.

FIG. 10 illustrates a four-state trellis structure 1000 used fordesigning codes for both BPSK and QPSK communication systems accordingto the invention. The four states of trellis structure 1000 are markedby the labels {C(X₁, X₂, 0), A₀, A₁}, {C(X₁, X₂, π), A₀, A₁}, {C(X₁, X₂,0), A₁, A₀}, and {C(X₁, X₂, π), A₁, A₀}. As will become more clearbelow, the label {C(X₁, X₂, 0), A₀, A₁} indicates that state 1 oftrellis 1000 is associated with space-time block code C(X₁, X₂, 0) andset partitions A₀ and A₁. The label {C(X₁, X₂, π), A₀, A₁} indicatesthat state 2 of trellis 1000 is associated with space-time block codeC(X₁, X₂, π) and set partitions A₀ and A₁. The label {C(X₁, X₂, 0), A₁,A₀} indicates that state 3 of trellis 1000 is associated with space-timeblock code C(X₁, X₂, 0) and set partitions A₁ and A₀. The label {C(X₁,X₂, π), A₁, A₀} indicates that state 4 of trellis 1000 is associatedwith space-time block code C(X₁, X₂, π) and set partitions A₁ and A₀.

As described herein, trellis structure 1000 is typically used with setpartitioning structures 700 and 800. Trellis structure 1000 isinterpreted in a manner similar to that described above for trellisstructure 600, as are the other trellis structures shown in FIGS. 11-14.

FIG. 11 illustrates a four-state trellis structure 1100 used fordesigning codes for 8-PSK communication systems according to theinvention. The four states of trellis structure 1100 are labeled {C(X₁,X₂, 0), A₀₀, A₀₁, A₁₀, A₁₁}, {C(X₁, X₂, π/2), A₀₀, A₀₁, A₁₀, A₁₁},{C(X₁, X₂, π), A₀₀, A₀₁, A₁₀, A₁₁}, and {C(X₁, X₂, 3π/2), A₀₀, A₀₁, A₁₀,A₁₁}. As described above, the label {C(X₁, X₂, 0), A₀₀, A₀₁, A₁₀, A₁₁}indicates that state 1 of trellis 1100 is associated with space-timeblock code C(X₁, X₂, 0) and set partitions A₀₀, A₀₁, A₁₀, and A₁₁. Thelabel {C(X₁, X₂, π/2), A₀₀, A₀₁, A₁₀, A₁₁} indicates that state 2 oftrellis 1100 is associated with space-time block code C(X₁, X₂, π/2) andset partitions A₀₀, A₀₁, A₁₀, and A₁₁. The label {C(X₁, X₂, π), A₀₀,A₀₁, A₁₀, A₁₁} indicates that state 3 of trellis 1100 is associated withspace-time block code C(X₁, X₂, π) and set partitions A₀₀, A₀₁, A₁₀, andA₁₁. The label {C(X₁, X₂, 3π/2), A₀₀, A₀₁, A₁₀, A₁₁} indicates thatstate 4 of trellis 1100 is associated with space-time block code C(X₁,X₂, 3π/2) and set partitions A₀₀, A₀₁, A₁₀, and A₁₁. Trellis structure1100 is typically used with set partitioning structure 900.

FIG. 12 illustrates a two-state trellis structure 1200 used fordesigning codes for BPSK and QPSK communication systems according to theinvention. The two states of trellis structure 1200 are labeled {C(X₁,X₂, 0), A₀, A₁} and {C(X₁, X₂, π), A₁, A₀}. The label {C(X₁, X₂, 0), A₀,A₁} indicates that state 1 of trellis 1200 is associated with space-timeblock code C(X₁, X₂, 0) and set partitions A₀ and A₁. The label {C(X₁,X₂, π), A₁, 0₁} indicates that state 2 of trellis 1200 is associatedwith space-time block code C(X₁, X₂, π) and set partitions A₁ and A₀.

As noted above, trellis structure 1200 is interpreted in a mannersimilar to that described above for trellis structure 600. Trellisstructure 1200 is typically used with set partitioning structures 700and 800.

FIG. 13 illustrates an eight-state trellis structure 1300 used fordesigning codes for 8-PSK communication systems according to theinvention. The eight states of trellis structure 1300 are labeled {C(X₁,X₂, 0), A₀₀, A₀₁, A₁₀, A₁₁} {C(X₁, X₂, π/2), A₀₀, A₀₁, A₁₀, A₁₁}, {C(X₁,X₂, π), A₀₀, A₀₁, A₁₀, A₁₁}, {C(X₁, X₂, π/2), A₀₀, A₀₁, A₁₀, A₁₁},{C(X₁, X₂, π), A₁₀, A₁₁, A₀₀, A₀₁, A₁₁}, {C(X₁, X₂, π/2), A₁₀, A₁₁, A₀₀,A₀₁}, {C(X₁, X₂, π), A₁₀, A₁₁, A₀₀, A₀₁}, and {C(X₁, X₂, 3π/2), A₁₀,A₁₁, A₀₀, A₀₁}. How to interpret the meaning of these labels will beunderstood by a person skilled in the relevant art given the descriptionherein. Trellis structure 1300 is typically used with set partitioningstructure 900.

FIG. 14 illustrates a four-state trellis structure 1400 used fordesigning codes for 8-PSK communication systems according to theinvention. The four states of trellis structure 1400 are labeled {C(X₁,X₂, 0), A₀₀, A₀₁}, {C(X₁, X₂, π), A₀₀, A₀₁}, {C(X₁, X₂, 0), A₁₀, A₀₀},and {C(X₁, X₂, π), A₁₀, A₀₀}. How to interpret the meaning of theselabels will be understood by a person skilled in the relevant art giventhe description herein. Trellis structure 1400 is also typically usedwith set partitioning structure 900. Trellis structure 1400 can be used,however, with any set partitioning structure according to the invention,as can the other trellis structures described herein.

Method for Designing Super-Orthogonal Space-Time Trellis Codes

In this section, a method 1700 for designing enhanced codes according tothe invention is described. Method 1700 is intended to be used fordesigning codes for diverse transmission communication systems. Method1700 is described with reference to the structures illustrated in FIGS.7-14, and with reference to the system embodiment features of theinvention illustrated in FIGS. 1-5.

Method 1700 starts at step 1710. In step 1710, a designer selects adiversity factor for the communications system to be implemented. Thisdiversity factor represents the number of transmission elements to beincluded in the transmitters of the system. In embodiments of theinvention, a diversity factor of two is selected. For example,transmitter 200 shown in FIG. 2 has two transmission elements orantennas 216 and 224. Similarly, transmitter 404 shown in FIG. 4 has twotransmission elements or optical transmitters 408 and 410. Asillustrated by transmitter 104 in FIG. 1, however, a diversity factorgreater than two may also be selected.

Often times, a designer will be asked to design a code for a preexistingcommunications system that uses transmitters having a set number oftransmission elements. In these situations, the designer may omit step1710 and start method 1700 at step 1720.

In step 1720, the designer selects a rate for the communications systemor code to be implemented. In embodiments of the invention, a rate of 1,2, or 3 bits/second/hertz is selected. The selected rate represents thenumber of bits transmitted in a given period of time.

Once the rate has been selected, other aspects of the communicationssystem and code are fixed. For example, a rate of 1 bit/second/hertzmeans that the system will have a constellation size of 2 (a BPSKsystem). A rate of 2 bits/second/hertz means the system will have aconstellation size of 4 (a QPSK system). A rate of 3 bits/second/hertzmeans that the system will have a constellation size of 8 (an 8-PSKsystem). In general, the constellation size (L) will equal 2^(b), whereb represents the selected rate. Also, as described herein, once the rateis selected, the number of input bits provided to lookup table 506 is2b. Thus, selecting a rate is an important design consideration.

In step 1730, the designer selects a number of states for the code. Inembodiments of the invention, the number of states selected is 2, 4, or8. Selecting a higher number of states for a given rate selectionimproves the performance of the code. For example, selecting a highernumber of states will improve coding gain distance (CGD). How todetermine CGD is described below. The CGD for set partitioningstructures 700, 800, and 900 are shown in FIGS. 7, 8 and 9,respectively.

In step 1740, the designer selects a trellis structure according to theinvention that has the number of states selected in step 1730. Asdescribed above, FIGS. 10-14 illustrates trellis structures that can beselected in this step of method 1700. The structure shown in FIGS. 10-14are referred to herein as super-orthogonal space-time trellisstructures. Trellis structure 1200 is a two-state trellis structure. Itcan be used, for example, to design both rate 1 bit/second/hertz andrate 2 bits/second/hertz communications systems according to theinvention. Trellis structure 1000 is a four-state trellis structure. Itcan be used, for example, to design both rate 1 bit/second/hertz andrate 2 bits/second/hertz communications systems according to theinvention. Trellis structure 1100 is another four-state trellisstructure. It can be used, for example, to design rate 3bits/second/hertz communications systems according to the invention.Trellis structure 1400 is also four-state trellis structure. It can beused, for example, to design rate 2.5 bits/second/hertz communicationssystems according to the invention. Trellis structure 1300 is aneight-state trellis structure. It can be used, for example, to designrate 3 bits/second/hertz communications systems according to theinvention. Trellis structures other than the ones described herein canbe designed in accordance with the invention, using the theory describedbelow, and selected in step 1740.

In step 1750, a set partitioning structure according to the invention isselected that corresponds to the rate selected in step 1720. Asdescribed above, FIGS. 7-9 illustrates set partitioning structuresaccording to the invention that may be selected in this step of method1700. The structure shown in FIGS. 7-9 are referred to herein assuper-orthogonal space-time code set partitioning structures. Setpartitioning structure 700 is used, for example, for designing rate 1bit/second/hertz codes. Set partitioning structure 800 is used, forexample, for designing rate 2 bits/second/hertz codes. Set partitioningstructure 900 is used, for example, for designing rate 3bits/second/hertz codes. As above, set partitioning structures otherthan the ones described herein can be designed in accordance with theinvention, using the theory described below, and selected in step 1750.

In step 1760, the trellis structure selected in step 1740 and the setpartitioning structure selected in step 1750 are used to assignparticular values to a set of transmission variable for each possiblecombination of input values. In order to illustrate this step, considerthe following description relating to the design and implementation of atwo-state, BPSK, super-orthogonal space-time trellis code according tothe invention.

As will be understood by a person skilled in the relevant communicationsart given the description herein, method 1700 and encoder 500B can beused to implement a two-state, rate 1 bit/second/hertz code for use bytransmitters having two transmission elements. These design constraintsrequire that the designer pick a diversity factor of 2 in step 1710, arate of 1 bit/second/hertz in step 1720, and 2 states for the code instep 1730. For this example, trellis structure 1200 is selected todesign the code (step 1740) in combination with set partitioningstructure 700 (step 1750). Trellis structure 1200 is a two-state trellisstructure that can be used to design both rate 1 bit/second/hertz andrate 2 bits/second/hertz communications systems according to theinvention. As described above, set partitioning structure 700 is usedfor designing rate 1 bit/second/hertz codes.

Selecting a rate of 1 bit/second/hertz means that 2 input bits areprovided to lookup table 506B of encoder 500B at time T₀. The number ofbits provided to lookup table 506B at time T₀ is dependent on the rateselected in step 1720. As described herein, one of these two input bitsis designated as state bit B_(A). The second input bit is designated asmodulation bit B_(B). In an embodiment, input bit B_(A) is used toselect transmission variable Θ, while input bit B_(B) is used to selectthe transmission variables X1 and X2. It makes no difference whether thefirst or the second bit entered into shift register 504B from input databit stream 502 is designated as state bit B_(A). Thus, for this example,the first of the two bits entered into shift register 504B is used as astate bit B_(A).

As illustrated in FIG. 5B, the output of lookup table 506B is threetransmission variables X1, X2, and Θ. Trellis structure 1200 will now beused in combination with set partitioning structure 700 to assignparticular values to transmission variable X1, X2, and Θ for eachpossible combination of input values.

The values assigned to transmission variable Θ are determined usingtrellis structure 1200. The value assigned to Θ for a particular set ofinputs is a function of the current state of encoder 500B at time T₀ andthe value of state bit B_(A) provided to lookup table 506B at time T₀.For example, if at time T₀, encoder 500B is in state 1 (i.e., at node1210) and state bit B_(A) equals 0, the value output by lookup table506B for Θ is 0. This will ensure that the encoder remains in state 1 asrequired by trellis structure 1200.

The value for Θ that is output by encoder 500B, for any set of inputs,can be determined using trellis structure 1200 if the state of encoder500 and the value of state bit B_(A) are known. Looking at trellisstructure 1200, one can see that a state bit B_(A) equal to 0 causes atransition from node 1210 (state 1) to node 1212 (state 1) along line1202. This is indicated by the label used for state 1 at node 1210. Thelabel {C(X₁, X₂, 0), A₀, A₁} indicates that a transition along line 1202is associated with partition subset A₀. This is because A₀ is the subsetlisted first after the function C(X₁, X₂, 0) (i.e., code associated withstate 1) and because line 1202 is the first line (from top to bottom oftrellis structure 1200) exiting node 1210. By looking at set partitionstructure 700 (or the subscript of A₀) one can see that a 0-bit (i.e.,state bit B_(A) being equal to 0) is required to move down the branchesof set partition structure 700 and arrive at subset A₀. From thecombination of information provided by trellis structure 1200 and setpartitioning structure 700, one is able to determine that, if at timeT₀, encoder 500B is in state 1 (i.e., at node 1210) and state bit B_(A)equals 0, the value output by lookup table 506B for Θ is 0 (i.e., thevalue for Θ) specified by the label for the state, after making thetransition indicated by the trellis structure 1200).

In the manner described above, one can use trellis structure 1200 andset partitioning structure 700 to determine the values of Θ that areoutput by lookup table 506B for a particular set of inputs. Inaccordance with trellis structure 1200, if encoder 500B is in state 1and state bit (B_(A)) equals 0 at time T₀, Θ is equal to 0. If encoder500B is in state 1 and state bit (B_(A)) equals 1 at time T₀, Θ is equalto π. If encoder 500B is in state 2 and state bit (B_(A)) equals 0 attime T₀, Θ) is equal to π. If encoder 500B is in state 2 and state bit(B_(A)) equals 1 at time T₀, Θ is equal to 0.

For this example, the values for the transmission variables X1 and X2are determined using trellis structure 1200 and set partitioningstructure 700. As described above, each transition on trellis structure1200 is associated with a particular subset of symbols identified at thebottom of set partitioning structure 700. For example, the label forstate 1 at node 1210, i.e., {C(X₁, X₂, 0), A₀, A₁}, indicates that atransition along line 1204 is associated with the partition subset A₁.This is because A₁ is listed second after the function C(X₁, X₂, 0) andbecause line 1204 is the second line (from top to bottom of trellisstructure 1200) exiting node 1210. Using trellis structure 1200, one candetermine that subset A₀ is associated with the transitions betweennodes 1210 and 1212 along line 1202 and between nodes 1214 and 1216along line 1208. Subset A₁ is associated with the transitions betweennodes 1210 and 1216 along line 1204 and between nodes 1214 and 1212along line 1206. As seen in FIG. 7, two groups of symbols 00 and 11 areassociated with the subset A₀ and two groups of symbols 01 and 10 areassociated with the subset A₁. One of the two groups of symbols in eachsubset is selected using modulation bit B_(B). In an embodiment, thegroup of bits selected are chosen from left to right so that whenmodulation bit B_(B) equals 0, group 00 or group 01 is selected, andwhen modulation bit B_(B) equals 1, group 11 or group 10 is selected.The first bit of each selected group is assigned to the transmissionvariable X1, and the second bit of each selected group is assigned tothe transmission variable X2.

Table 1 below can be constructed using trellis structure 1200 and setpartitioning structure 700. Table 1 represents the encoding for thetransmission variables of a two-state, 1 bit/second/hertzsuper-orthogonal space-time trellis code according to the invention. Thestructure of Table 1 can be implemented as lookup table 506B in encoder500B. TABLE 1 Input Variables Output Variables State B_(A) B_(B) X1 X2 Θ1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 π 1 1 1 1 0 π 2 0 0 0 0 π 2 0 1 1 1 π2 1 0 0 1 0 2 1 1 1 0 0

In accordance with method 1700, Table 2 can be constructed using trellisstructure 1200 and set partitioning structure 800. Table 2 representsthe encoding for the transmission variables of a two-state, 2bits/second/hertz super-orthogonal space-time trellis code according tothe invention. The structure of Table 2 can also be implemented aslookup table 506B in encoder 500B. TABLE 2 Input Variables OutputVariables State B_(A) B_(B) X1 X2 Θ 1 0 000 0 0 0 1 0 001 2 2 0 1 0 0100 2 0 1 0 011 2 0 0 1 0 100 1 1 0 1 0 101 3 3 0 1 0 110 1 3 0 1 0 111 31 0 1 1 000 0 1 π 1 1 001 2 3 π 1 1 010 0 3 π 1 1 011 2 1 π 1 1 100 1 0π 1 1 101 3 2 π 1 1 110 1 2 π 1 1 111 3 0 π 2 1 000 0 1 π 2 1 001 2 3 π2 1 010 0 3 π 2 1 011 2 1 π 2 1 100 1 0 0 2 1 101 3 2 0 2 1 110 1 2 0 21 111 3 0 0 2 0 000 0 0 π 2 0 001 2 2 π 2 0 010 0 2 π 2 0 011 2 0 π 2 0100 1 1 π 2 0 101 3 3 π 2 0 110 1 3 π 2 0 111 3 1 π

In accordance with method 1700, Table 3 can be constructed using trellisstructure 1300 and set partitioning structure 900. Table 3 representsthe partial encoding for the transmission variables of a four-state, 3bits/second/hertz super-orthogonal space-time trellis code according tothe invention. In particular, Table 3 shows the encoding for the inputsthat cause a transition from state 1 to state 1 and from state 1 tostate 2. Table 3 illustrates the use of trellis structure 1300 and setpartitioning structure 900 for designing a code according to theinvention. TABLE 3 Input Variables Output Variables State B_(A) B_(B) X1X2 Θ 1 10 0000 0 1 0 1 10 0001 4 5 0 1 10 0010 0 5 0 1 10 0011 4 1 0 110 0100 2 3 0 1 10 0101 6 7 0 1 10 0110 2 7 0 1 10 0111 6 3 0 1 10 10000 3 0 1 10 1001 4 7 0 1 10 1010 0 7 0 1 10 1011 4 3 0 1 10 1100 2 1 0 110 1101 6 5 0 1 10 1110 2 5 0 1 10 1111 6 1 0 1 11 0000 1 0 π/2 1 110001 5 4 π/2 1 11 0010 1 4 π/2 1 11 0011 5 0 π/2 1 11 0100 3 2 π/2 1 110101 7 6 π/2 1 11 0110 3 6 π/2 1 11 0111 7 2 π/2 1 11 1000 1 2 π/2 1 111001 5 6 π/2 1 11 1010 1 6 π/2 1 11 1011 5 2 π/2 1 11 1100 3 0 π/2 1 111101 7 4 π/2 1 11 1110 3 4 π/2 1 11 1111 7 0 π/2

As will be understood by a person skilled in the relevant communicationsart given the description herein, method 1700 and the structuresdescribed herein can be used to systematically design a variety ofsuper-orthogonal space-time trellis codes having full diversity andenhanced coding gain.

Detailed Theory of the Invention

In this section, the detailed theory of super-orthogonal space-timetrellis codes is described.

Overview

An example of a full-rate, full-diversity complex space-time block codeis a scheme proposed by Alamouti. (See S. M. Alamouti, “A simpletransmitter diversity scheme for wireless communications,” in IEEEJournal on Selected Areas of Communications, Vol. 16, pages 1451-1458(November 1998), which is incorporated herein by reference in itsentirety.) Alamouti's scheme is defined by the following transmissionmatrix: $\begin{matrix}{{C\quad\left( {x_{1},x_{2}} \right)} = {\begin{pmatrix}x_{1} & x_{2} \\{- x_{2}^{*}} & x_{1}^{*}\end{pmatrix}.}} & (1)\end{matrix}$The proposed scheme can be used for N=2 transmit antennas and any numberof receive antennas. It transmits 2b bits every two symbol intervals,where the 2D constellation size is L=2^(b). 2b bits arrive at an encoderand the encoder chooses 2 modulation symbols s₁ and s₂. Using C(s₁, s₂),the encoder transmits s₁ from antenna one and s₂ from antenna two attime one. Then, the encoder transmits −s₂* from antenna one and −s₁*from antenna two at time two. The next 2b bits to arrive at the encoderdetermine what is transmitted at times three and four in a similar way.While this scheme provides diversity gain, it does not provideadditional coding gain.

By concatenating an outer trellis code designed for an additive whiteGaussian noise (AWGN) channel with a space-time block code, however,additional performance gain is obtained. To see this, consider each ofthe 2^(2b) orthogonal matrices generated by the space-time block code ofEQ. (1) (i.e., Equation 1) as a 4D signal point (strictly speaking it isnot 4D). The outer trellis code's task is to select on of the 4D signalpoints to be transmitted based on the current state and the 2b inputbits. It has been shown that for a slow fading channel, the trellis codeshould be based on the set partitioning concepts of Ungerboeck codes forthe AWGN channel. (See S. Alamouti et al., “Trellis-coded modulation andtransmit diversity: design criteria and performance evaluation,” IEEEInternational Conference on Universal Personal Communications(ICUPC-98), Vol. 2, pages 917-920 (1998), which is incorporated hereinby reference in its entirety.) A shortcoming of this proposed scheme isthat there is a rate loss associated with achieving any coding gain ifthe constituent 2D signal constellation size does not increase.Alternatively for the same rate, the basic 2D signal constellation sizehas to increase resulting in increased peak-to-average ratio. This isbecause neither this scheme nor the scheme of EQ. (1) are using all ofthe possible 4D constellations.

To elaborate, consider other codes that provide behaviors similar tothose of EQ. (1) for the same rate and number of transmit antennas. Theset of all such codes which only use x₁, x₂, and their conjugates withpositive or negative signs are listed below: $\begin{pmatrix}x_{1} & x_{2} \\x_{2}^{*} & {- x_{1}^{*}}\end{pmatrix},\begin{pmatrix}{- x_{1}} & x_{2} \\x_{2}^{*} & x_{1}^{*}\end{pmatrix},\begin{pmatrix}x_{1} & {- x_{2}} \\x_{2}^{*} & x_{1}^{*}\end{pmatrix},\begin{pmatrix}x_{1} & x_{2} \\{- x_{2}^{*}} & x_{1}^{*}\end{pmatrix},\begin{pmatrix}{- x_{1}} & {- x_{2}} \\x_{2}^{*} & {- x_{1}^{*}}\end{pmatrix},\begin{pmatrix}{- x_{1}} & x_{2} \\{- x_{2}^{*}} & {- x_{1}^{*}}\end{pmatrix},\begin{pmatrix}x_{1} & {- x_{2}} \\{- x_{2}^{*}} & x_{1}^{*}\end{pmatrix},{\begin{pmatrix}{- x_{1}} & {- x_{2}} \\{- x_{2}^{*}} & x_{1}^{*}\end{pmatrix}.}$The union of all these codes is referred to herein as “super-orthogonalcode” set C. Using just one of the constituent codes from C (e.g. thecode in EQ. (1)) one cannot create all possible orthogonal 2×2 matricesfor a given constellation. To make this point more clear, consider aBPSK constellation. It can be shown that one can build all possible 2×2orthogonal matrices using two of the codes in C. For example, one cangenerate the following four 2×2 matrices using the code in EQ. (1):$\begin{matrix}{\begin{pmatrix}1 & 1 \\{- 1} & 1\end{pmatrix},\begin{pmatrix}{- 1} & {- 1} \\1 & {- 1}\end{pmatrix},\begin{pmatrix}{- 1} & 1 \\{- 1} & {- 1}\end{pmatrix},{\begin{pmatrix}1 & {- 1} \\1 & 1\end{pmatrix}.}} & (2)\end{matrix}$There are four other possible distinct orthogonal 2×2 matrices, whichare listed below: $\begin{matrix}{\begin{pmatrix}{- 1} & 1 \\1 & 1\end{pmatrix},\begin{pmatrix}1 & {- 1} \\{- 1} & {- 1}\end{pmatrix},\begin{pmatrix}1 & 1 \\1 & {- 1}\end{pmatrix},{\begin{pmatrix}{- 1} & {- 1} \\{- 1} & 1\end{pmatrix}.}} & (3)\end{matrix}$To create these additional matrices, one can use the following code fromthe set C: $\begin{matrix}{\begin{pmatrix}{- x_{1}} & x_{2} \\x_{2}^{*} & x_{1}^{*}\end{pmatrix}.} & (4)\end{matrix}$A set that includes all 2×2 orthogonal matrices from (2) and (3) isreferred to herein as set S₂. It is important to note that the rank of amatrix (which determines diversity) based on the difference between anytwo distinct matrices within either (2) or (3) is 2, but the rank of amatrix obtained by considering the difference between any two elementsin (2) and (3) is 1.

Parameterized Class of Space-Time Block Codes

In accordance with the present invention, the following class oforthogonal designs is used as transmission matrices: $\begin{matrix}{{C\quad\left( {x_{1},x_{2},\theta} \right)} = {\begin{pmatrix}{x_{1}{\mathbb{e}}^{\quad{j\theta}}} & x_{2} \\{{- x_{2}^{*}}{\mathbb{e}}^{j\theta}} & x_{1}^{*}\end{pmatrix}.}} & \left( {5A} \right)\end{matrix}$Note that θ=0 provides the code in EQ. (1). An objective of theinvention is to use L-PSK constellation sets (i.e., PSK constellationscontaining L points) and design codes that do not expand theconstellation signals. Since the transmitted signals are from a PSKconstellation, ${\mathbb{e}}^{\frac{{j2\pi}\quad d}{L}},{l = 0},$1, . . . , L, the peak-to-average power ratio of the transmitted signalsis equal to one. Therefore, there is no need for an amplifier thatprovides a higher linear operation region. In order to achieve this, theinvention uses θ=2π/′/L, where l′=0, 1, . . . , L−1. For example, theinvention uses θ={0, π} and θ={0, π/2, π, 3π/2} for BPSK and QPSK,respectively. By using C(x₁, x₂, 0) and C(x₁, x₂, π) for BPSKconstellations, one can generate all 2×2 orthogonal matrices in S₂. Infact, C(x₁, x₂, 0) is the code in (1) and C(x₁, x₂, π) is the code in(4).

The invention is not limited to using the class of orthogonal designsindicated by EQ. (5A). For example, another class of designs that can beused as transmission matrices in accordance with the invention is:$\begin{matrix}{{C\quad\left( {x_{1},x_{2},\theta} \right)} = \begin{pmatrix}{x_{1}{\mathbb{e}}^{\quad{j\theta}}} & {x_{2}{\mathbb{e}}^{j\theta}} \\{- x_{2}^{*}} & x_{1}^{*}\end{pmatrix}} & \left( {5B} \right)\end{matrix}$Other designs will be known and understood by persons skilled in therelevant communications art given the description herein.

Set Partitioning

Following the definitions in Tarokh et al. for a full-diversity code(see V. Tarokh et al., “Space-time codes for high data rate wirelesscommunications: Performance analysis and code construction,” in IEEETrans. Inform. Theory, Vol. 44, pages 744-765 (March 1998), which isincorporated herein by reference in its entirety), the minimum of thedeterminant of the matrix A(c₁, c₂)=B(c₁, c₂)B*(c₁, c₂) over allpossible pairs of distinct codewords c₁ and c₂ corresponds to the codinggain. The coding gain distance (CGD) between codewords c₁ and c₂ isdefined herein as d²(c₁, c₂)=|A(c₁, c₂)| (in general if instead of afull-diversity, N, we have a code with diversity r<N, the distance canbe defined as the r^(th) root of the sum of the determinants of all r×rprincipal cofactors of A(c₁, c₂)). CGD is used herein, instead ofEuclidean distance, to define a set partitioning similar to Ungerboeck'sset partitioning. (See G. Ungerboeck, “Channel coding formultilevel/phase signals,” in IEEE Trans. Inform. Theory Vol. 28, pages55-67 (January 1982), which is incorporated herein by reference in itsentirety.)

Consider now a case where one utilizes only the code of EQ. (1). One canuse a multiple trellis-coded modulation (MTCM) scheme (see D. Divsalaxet al., “Multiple trellis coded modulation (MTCM),” IEEE Trans.Communications, Vol. 36, pages 410-419 (April 1988), which isincorporated herein by reference in its entirety) and assign the code inEQ. (1) with two specific constellation symbols to each trellis path.(See S. Alamouti et al., “Trellis-coded modulation and transmitdiversity: design criteria and performance evaluation,” IEEEInternational Conference on Universal Personal Communications(ICUPC-98), Vol. 2, pages 917-920 (1998)). Assuming one uses a PSKconstellation with L signals, each signal can be represented by${s = {\mathbb{e}}^{\frac{{j2\pi}\quad d}{L}}},$l=0, 1, . . . , L or s=e^(jlω), where$\omega = {\frac{{2\pi}\quad}{L}.}$Now consider two distinct sets of constellation symbols (s₁ ¹=e^(jk) ¹^(ω),s₂ ¹=e^(jl) ¹ ^(ω)) and (s₁ ²=e^(jk) ² ^(ω),s₂ ²=e^(jl) ² ^(ω)) tocalculate B and A. For parallel transitions in a trellis, B and A aregiven by EQs. (6) and (7), respectively. $\begin{matrix}{{B = \begin{pmatrix}{\mathbb{e}}^{j\quad k_{1}\omega} & {- {\mathbb{e}}^{j\quad k_{2}\omega}} & {\mathbb{e}}^{j\quad l_{1}\omega} & {- {\mathbb{e}}^{j\quad l_{2}\omega}} \\{\mathbb{e}}^{{- j}\quad l_{2}\omega} & {- {\mathbb{e}}^{{- j}\quad l_{1}\omega}} & {\mathbb{e}}^{{- j}\quad k_{1}\omega} & {- {\mathbb{e}}^{{- j}\quad k_{2}\omega}}\end{pmatrix}},} & (6) \\\begin{matrix}{A = {BB}^{*}} \\{= {\begin{pmatrix}{4 - {2{\cos\quad\left\lbrack {\omega\quad\left( {k_{2} - k_{1}} \right)} \right\rbrack}} - {2{\cos\quad\left\lbrack {\omega\quad\left( {l_{2} - l_{1}} \right)} \right\rbrack}}} & 0 \\0 & {4 - {2{\cos\quad\left\lbrack {\omega\quad\left( {k_{2} - k_{1}} \right)} \right\rbrack}} - {2{\cos\quad\left\lbrack {\omega\quad\left( {l_{2} - l_{1}} \right)} \right\rbrack}}}\end{pmatrix}.}}\end{matrix} & (7)\end{matrix}$

Using EQ. (7), one can show that|A|={4−2 cos [ω(k ₂ −k ₁)]−2 cos [ω(l ₂ −l ₁)]}².  (8)Now, if there are two codewords which differ in P sets of constellationsymbols, it can be shown that A₁₂=A₂₁=0. Also, if for the firstcodeword, one denotes the set of constellation symbols by (s₁ ¹,s₂¹)^(p)=(e^(jk) ^(l) ^(p) ^(ω),e^(jl) ¹ ^(p) ^(ω)), p=0, 1, . . . , P andfor the second codeword, one denotes the set of constellation symbols by(s₁ ²,s₂ ²)^(p)=(e^(jk) ² ^(p) ^(ω),e^(jl) ² ^(p) ^(ω)) p=0, 1, . . . ,P, then $\begin{matrix}{{A} = \left\{ {{\sum\limits_{p = 1}^{P}\quad 4} - {2{\cos\quad\left\lbrack {\omega\quad\left( {k_{2}^{p} - k_{1}^{p}} \right)} \right\rbrack}} - {2{\cos\quad\left\lbrack {\omega\quad\left( {l_{2}^{p} - l_{1}^{p}} \right)} \right\rbrack}}} \right\}^{2}} & (9)\end{matrix}$Note that each term in the sum is non-negative and therefore$\begin{matrix}\begin{matrix}{{A} = {\left\{ \quad\begin{matrix}{{\sum\limits_{p\quad = \quad 1}^{\quad P}\quad 4} - {2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{k_{\quad 2}^{\quad p}\quad - \quad k_{\quad 1}^{\quad p}} \right)} \right\rbrack}}\quad -} \\{\quad{2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{l_{\quad 2}^{\quad p}\quad - \quad l_{\quad 1}^{\quad p}} \right)} \right\rbrack}}}\end{matrix} \right\}^{2} \geq}} \\{\sum\limits_{p\quad = \quad 1}^{\quad P}\quad{\left\{ {4\quad - \quad{2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{k_{\quad 2}^{\quad p}\quad - \quad k_{\quad 1}^{\quad p}} \right)} \right\rbrack}}\quad - \quad{2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{l_{\quad 2}^{\quad p}\quad - \quad l_{\quad 1}^{\quad p}} \right)} \right\rbrack}}} \right\}^{2}.}}\end{matrix} & (10)\end{matrix}$The performance of this code depends on the choice of trellis,constellation set, and set partitioning. As an example, consider thefour-state trellis of FIG. 6. If one uses the code of EQ. (1) and anL-PSK constellation, one can generate 2^(2b) orthogonal 2×2 matrices(L=2^(b)). Dividing these matrices into four sets provides a rate b/2bits/sec/Hz space-time trellis code. This is similar to the secondAlamouti scheme, noted above, which can only transmit a rate that ishalf of the maximum possible rate. (See V. Tarokh, et al., “Space-timecodes for high data rate wireless communications: performance analysisand code construction,” IEEE Trans. Inform. Theory, Vol. 44, pages744-765 (March 1998), which is incorporated herein by reference in itsentirety.) Based on the coding distances calculated in EQs. (8) and (9),one can show that the coding gain of such a space-time trellis code isdominated by parallel transitions.

The optimal set partitioning for BPSK, QPSK, and 8-PSK are illustratedin FIGS. 7, 8, and 9, respectively. As can be seen from these Figures,the minimum CGD increases (or remains the same) as one goes down eachlevel in the tree. As described herein, the branches at each level canbe used to design a trellis code with a specific rate. This setpartitioning, in addition to the known rules for designing an MTCM,provides a class of space-time trellis codes. This is a differentapproach for designing the codes then that presented by Alamouti (i.e.,using an MTCM scheme concatenated with the Alamouti code).

Super-Orthogonal Space-Time Trellis Codes

FIGS. 10-14 illustrate examples of super-orthogonal space-time trelliscodes according to the invention. In these figures, C(x₁, x₂, 0)represents the particular member of a parameterized space-time blockcode which is used at a specific state. The corresponding sets from theset partitioning structures of FIGS. 7-9 are shown next to each state.FIG. 10 shows an example of a super-orthogonal space-time trellis code.In this example, one can use BPSK and the corresponding set partitioningin FIG. 7. C(x₁, x₂, 0) is used when departing from states zero and two,and C(x₁, x₂, π) is used when departing from states one and three. Notethat, with this structure, one has eight possible orthogonal 2×2matrices instead of four which allows one to design a full-rate code.The CGD of this code is 64 which can be found in FIG. 7 and Table 4. Innext section, it is shown that parallel transitions are dominant incalculating the CGD for this code.

Codes with different number of states and at different rates can besystematically designed using the set partitioning in FIGS. 7-9 byapplying the general well-known rules defined in Ungerboeck and Divsalaxto design MTCM schemes. (See G. Ungerboeck, “Channel coding formultilevel/phase signals,” IEEE Trans. Inform. Theory, Vol. 28, pages55-67 (January 1982); and D. Divsalax et al., “Multiple trellis codedmodulation (MTMC),” IEEE Trans. Communications, Vol. 36, pages 410-419(April 1988), both of which are incorporated herein by reference intheir entirety.) While the examples herein focus on the design offull-rate codes, in general, codes with lower rates can be designed toprovide higher CGDs. The trellis structure of FIG. 14 is one example ofa 4-state, rate 2.5 bits/sec/Hz code using 8-PSK with a CGD of 4. Basedon the description herein, a person skilled in the relevant art willunderstand how to use the invention to design other lower rate codesthat provided higher CGDs.

Table 4 below tabulates the CGDs of the super-orthogonal space-timetrellis codes described herein, and compares them to corresponding codesfrom Tarokh. (See V. Tarokh, et al., “Space-time codes for high datarate wireless communications: performance analysis and codeconstruction,” IEEE Trans. Inform. Theory, Vol. 44, pages 744-765 (March1998)).

CGD Analysis

In this section, the coding gain of different super-orthogonalspace-time trellis codes according to the invention are derived. Themain results in this section, which provide the dominant path for theCGD calculation in the trellis, depend on the choice of the trellis. Theconstellation would only affect the final value of the CGD. The approachherein is general enough to be extended to other trellises in theliterature.

Consider the trellis in FIG. 6, which corresponds to the codes of FIG.10. Note that when there are parallel transitions between two states. Adifferent path is assigned to each possible constellation matrix(symbols), which is defined by the set partitioning in FIGS. 7-9. Twocodewords may only differ in P=1 set of constellation symbols. However,due to the structure of the trellis, it is impossible to have twocodewords which differ in two sets of constellation symbols (P=2).Because, for example, if two codewords diverge from state zero, theyhave to go through at least 3 paths to reconverge. Therefore, theshortest value of P excluding parallel transitions is three. Thefollowing Lemma is used to calculate the CGD.

Lemma 1: For the codes of FIG. 10, the minimum value of the CGD when P=3is greater than the minimum value of the CGD when P=1. Proof. Withoutloss of generality one can assume two codewords diverging from statezero and reconverging after P paths to state zero. For paralleltransitions (P=1), one can calculate the CGD from EQ. (8). In this case,min |A|=64 for the rate 1 bit/second/hertz code of FIGS. 10 and 16 forthe rate 2 bits/second/hertz code of FIG. 10. For P=3, consider atypical case where the first codeword stays at state zero. For thesecond codeword, the first and third transitions (diverging andconverging to state zero) use C(x₁, x₂, 0) and the second transitionuses C(x₁, x₂, π). It can be shown that $\begin{matrix}\begin{matrix}{{A} = \left\{ {4 - {2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{k_{\quad 2}^{1} - k_{\quad 1}^{1}} \right)} \right\rbrack}} - {2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{l_{\quad 2}^{\quad 1} - l_{\quad 1}^{\quad 1}} \right)} \right\rbrack}} +} \right.} \\{4 - {2\quad{\cos\quad\left\lbrack {k_{\quad 2}^{3} - k_{\quad 1}^{3}} \right\rbrack}} - {2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{l_{\quad 2}^{\quad 3} - l_{\quad 1}^{\quad 3}} \right)} \right\rbrack}} +} \\\left. {4 + {2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{k_{\quad 2}^{2} - k_{\quad 1}^{2}} \right)} \right\rbrack}} - {2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{l_{\quad 2}^{\quad 2} - l_{\quad 1}^{\quad 2}} \right)} \right\rbrack}}} \right\} \\{\left\{ {4 - {2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{k_{\quad 2}^{1} - k_{\quad 1}^{1}} \right)} \right\rbrack}} - {2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{l_{\quad 2}^{\quad 1} - l_{\quad 1}^{\quad 1}} \right)} \right\rbrack}} +} \right.} \\{4 - {2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{k_{\quad 2}^{3} - k_{\quad 1}^{3}} \right)} \right\rbrack}} - {2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{l_{\quad 2}^{\quad 3} - l_{\quad 1}^{\quad 3}} \right)} \right\rbrack}} +} \\{4 - {2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{k_{\quad 2}^{2} - k_{\quad 1}^{2}} \right)} \right\rbrack}} + {2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{l_{\quad 2}^{\quad 2} - l_{\quad 1}^{\quad 2}} \right)} \right\rbrack}} -} \\{8\quad{\left( {1 + {\cos\quad\left\lbrack {\omega\quad\left( {k_{2}^{2} - k_{1}^{2} + l_{1}^{2} - l_{2}^{2}} \right)} \right\rbrack}} \right).}}\end{matrix} & (11)\end{matrix}$If one definesa=4−2 cos [ω(k ₂ ¹ −k ₁ ¹)]−2 cos [ω(l ₂ ¹ −l ₁ ¹)],c=4−2 cos [ω(k ₂ ³ −k ₁ ³)]−2 cos [ω(l ₂ ³ −l ₁ ³)],b ₁=4+2 cos [ω(k ₂ ² −k ₁ ²)]−2 cos [ω(l ₂ ² −l ₁ ²)],b ₂=4−2 cos [ω(k ₂ ² −k ₁ ²)]+2 cos [ω(l ₂ ² −l ₁ ²)],d=8(1+cos [ω(k ₂ ² k ₁ ² +l ₁ ² −l ₂ ²)]),then|A|=(a+b ₁ +c)(a+b ₂ +c)−d,  (12)where a, b₁, b₂, c, d≧0.This givesmin |A|≧(min a+min b ₁+min c)(min a+min b ₂+min c)−max d.  (13)For the 1 bit/second/hertz code of FIG. 10, min a=min c=4, min b₁=minb₂=4, and max d=16. Therefore,min |A|≧128  (14)Also, for k₂ ¹=k₁ ¹=l₁ ¹=k₂ ²−k₁ ²=l₁ ²=l₂ ²=k₂ ³=k₁ ³=l₁ ³=0 and l₂¹=l₂ ³=1 we have |A|=128 which meansmin |A|≦128  (15)Combining inequalities (14) and (15) providesmin |A|=128  (16)which is greater than 64.

For the 2 bits/second/hertz code of FIG. 10, min a=min c=2, min b₁=minb₂=4, and max d=16. Therefore,min |A|≧48  (17)which is greater than 16.

Corollary 1: The coding gain of the 1 bit/second/hertz code of FIG. 10is 8.

Proof. It is easy to show that for any two codewords that differ in morethan 3 paths (P>3) there are three transitions similar to the ones inthe proof of Lemma IV.1. Also, calculating |A| shows that the extratransitions can only add to the CGD positively. Therefore, the minimumvalue of the CGD when P>3 is greater than the minimum value of the CGDwhen P=3. This proves that the minimum CGD for the 1 bit/second/hertzcode of FIG. 10 is dominated by parallel transitions and is equal to 64.Therefore, the coding gain is 8.

Corollary 2: The coding gain of the 2 bits/second/hertz code of FIG. 10is 4.

Proof. Using the same argument as the one in the proof of lastcorollary, the minimum CGD for the code is dominated by paralleltransitions and is equal to 16. Therefore, the coding gain is 4.

Note that specific codes are considered herein for clarity ofpresentation. One can show similar results for any trellis for which itis impossible to diverge from a state and reconverge to the same statein P=2 transitions. For the trellises which allow such a transition,consider the following Lemma.

Lemma 2: For the codes of FIGS. 11 and 13, the minimum value of the CGDwhen P=1 is greater than the minimum value of the CGD when P=2.Proof: Without loss of generality one can assume two codewords divergingfrom state zero and reconverging after P paths to state zero. Forparallel transitions (P=1), one can calculate the CGD from EQ. (8). Inthis case, min |A|=4. For P=2, consider a typical case where the firstcodeword stays at state zero. For the second codeword, the firsttransition diverging from state zero uses C(x₁, x₂, 0) and the secondtransition converging to state zero uses C(x₁, x₂, θ), θ=π/2, π, 3π/2.It can be shown that $\begin{matrix}\begin{matrix}{{A} = \left\{ {4 - {2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{k_{\quad 2}^{1} - k_{\quad 1}^{1}} \right)} \right\rbrack}} - {2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{l_{\quad 2}^{\quad 1} - l_{\quad 1}^{\quad 1}} \right)} \right\rbrack}} +} \right.} \\\left. {4 - {2\quad{\cos\quad\left\lbrack {\omega\quad\left( {k_{\quad 2}^{2} - k_{\quad 1}^{2}} \right)} \right\rbrack}} - {2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{l_{\quad 2}^{\quad 2} - l_{\quad 1}^{\quad 2}} \right)} \right\rbrack}}} \right\} \\{\left\{ {4 - {2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{k_{\quad 2}^{1} - k_{\quad 1}^{1}} \right)} \right\rbrack}} - {2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{l_{\quad 2}^{\quad 1} - l_{\quad 1}^{\quad 1}} \right)} \right\rbrack}} +} \right.} \\{\left. {4 - {2\quad{\cos\quad\left\lbrack {{\omega\quad\left( \quad{k_{\quad 2}^{2} - k_{\quad 1}^{2}} \right)} - \theta} \right\rbrack}} + {2\quad{\cos\quad\left\lbrack {\omega\quad\left( \quad{l_{\quad 2}^{\quad 2} - l_{\quad 1}^{\quad 2}} \right)} \right\rbrack}} - \theta} \right\} -} \\{\left( {2 - {2\cos\quad\theta}} \right){\left( {2 - {2{\cos\quad\left\lbrack {\omega\quad\left( {k_{2}^{2} - k_{1}^{2} + l_{1}^{2} - l_{2}^{2}} \right)} \right\rbrack}}} \right).}}\end{matrix} & (20)\end{matrix}$If one definesa=4−2 cos [ω(k ₂ ¹ −k ₁ ¹)]−2 cos [ω(l ₂ ¹ −l ₁ ¹)],b ₁=4−2 cos [ω(k ₂ ² −k ₁ ²)]−2 cos [ω(l ₂ ² −l ₁ ²)],b ₂=4−2 cos [ω(k ₂ ² −k ₁ ²)−θ]+2 cos [ω(l ₂ ² −l ₁ ²)−θ],d=(2−2 cos θ)(2−2 cos [ω(k ₂ ² −k ₁ ² +l ₁ ² −l ₂ ²)]),then|A|=(a+b ₁)(a+b ₂)−d=a ² +a(b ₁ +b ₂)+b₁ b ₂ −d,  (21)where a, b₁, b₂, d≧0. This givesmin |A|=(min a)²+min[(min a)(b ₁ +b ₂)+b₁ b ₂ −d].  (22)This is due to the fact that a depends on (k₂ ¹,k₁ ¹,l₂ ¹,l₁ ¹) whileb₁, b₂, and d are independent of (k₂ ¹,k₁ ¹,l₂ ¹,l₁ ¹) Since mina=0.5858, one hasmin |A|=0.3431+min[0.5858(b ₁ +b ₂)+b ₁ b ₂ −d]=2.54<4.  (23)

Corollary 3: The coding gain of the codes of FIGS. 11 and 13 is 1.59.

Proof: It is easy to show that for any two codewords that differ in morethan 2 paths (P>2) there are two transitions similar to the ones in theproof of Lemma IV.2. Also, calculating |A| shows that the extratransitions can only add to the CGD positively. This proves that theminimum CGD is dominated by paths with P=2 transitions and is equal to2.54. Therefore, the coding gain is 1.59. TABLE 4 CGD in (Tarokh et al.,IEEE No. of rate Trans. Inform. Theory Figure states (bits/sec/Hz) CGD44: 744-765 (1998)) 10 4 1 64 — 10 4 2 16 4 11 4 3 2.54 — 12 2 1 48 — 122 2 12 — 13 8 3 2.54 2 14 4 2.5 4 —

Simulation Results

In this section, simulation results for super-orthogonal space-timetrellis codes (SOSTTCs), using two transmit antennas and one receiveantenna, are provided. These results are compared to results of similarcorresponding space-time trellis codes (STTCs) from Tarokh, when acomparable code exists. (See V. Tarokh, et al., “Space-time codes forhigh data rate wireless communications: performance analysis and codeconstruction,” IEEE Trans. Inform. Theory, Vol. 44, pages 744-765 (March1998)). In all simulations, similar to the results in Tarokh, a frameconsists of 130 transmissions out of each transmit antenna.

FIG. 15 shows the frame error probability results versus signal-to-noiseratio (SNR) for the 1 bit/second/hertz codes in FIGS. 10 and 12. Both ofthese codes are full-rate and transmit 1 bit/second/hertz using a BPSKconstellation. Note that one cannot design such a code using the schemeof Alamouti. (See S. Alamouti et al., “Trellis-coded modulation andtransmit diversity: design criteria and performance evaluation,” IEEEInternational Conference on Universal Personal Communications(ICUPC-98), Vol. 2, pages 917-920 (1998).)

FIG. 16 shows the simulation results for transmitting 2bits/second/hertz using a QPSK constellation. The codes of FIGS. 10 and12 are denoted by “SOSTTC 4 states” and “SOSTTC 2 states” respectively.The corresponding results for a code with the same rate and 4 statesfrom Tarokh is also provided (“STTC”) for comparison. (See V. Tarokh, etal., “Space-time codes for high data rate wireless communications:performance analysis and code construction,” IEEE Trans. Inform. Theory,Vol. 44, pages 744-765 (March 1998).) As can be seen from FIG. 16, the4-state super-orthogonal space-time trellis code according to theinvention outperforms the corresponding space-time trellis code by morethan 2 dB. The performance of the 4-state super-orthogonal space-timetrellis code according to the invention is very close to that of a64-state space-time trellis code.

Summary

In this theory section, results for communications systems using twotransmit antennas have been presented. The theory, however, is generaland can be extended to more than two transmit antennas. For thispurpose, one can use the general space-time block codes introduced inTarokh. (See V. Tarokh et al., “Space-time block codes from orthogonaldesigns,” IEEE Trans. Inform. Theory, Vol. 45, pages 1456-1467 (July1999), which is incorporated herein by reference in its entirety.) Thecode design strategy described herein is general enough to be used witha quasi-orthogonal space-time block code or any other structure whichguarantees diversity. (See H. Jafaxkhani, “A quasi-orthogonal space-timeblock code,” IEEE Trans. Communications, Vol. 49, pages 1-4 (January2001), which is incorporated herein by reference in its entirety.)

CONCLUSION

Various embodiments of the present invention have been described above.It should be understood that these embodiments have been presented byway of example only, and not limitation. It will be understood by thoseskilled in the relevant art that various changes in form and details ofthe embodiments described above may be made without departing from thespirit and scope of the present invention as defined in the claims.Thus, the breadth and scope of the present invention should not belimited by any of the above-described exemplary embodiments, but shouldbe defined only in accordance with the following claims and theirequivalents.

1. A telecommunications transmitter, comprising: a first module that receives a plurality of input data bits; a second module, coupled to the first module, that generates transmission variables X₁, X₂, and θ based on the plurality of input data bits; and a third module that output a first signal X₁e^(jθ) and a second signal X₂ at a first predetermined time, and that outputs a third signal −X₂*e^(jθ) and a fourth signal X₁* at a second predetermined time.
 2. The transmitter of claim 1, wherein the first, second, third, and fourth signals form part of a multipoint PSK constellation.
 3. The transmitter of claim 1, wherein the peak-to-average power ration of transmitted signals is equal to one.
 4. The transmitter of claim 1, wherein θ is selected from the group 0, π/2, π, and 3π/2.
 5. A telecommunications transmitter, comprising: a first module that receives a plurality of input data bits; a second module, coupled to the first module, that generates transmission variables X₁, X₂, and θ based on the plurality of input data bits; and a third module that output a first signal X₁e^(jθ) and a second signal X₂e^(jθ) at a first predetermined time, and that outputs a third signal −X₂* and a fourth signal X₁* at a second predetermined time.
 6. The transmitter of claim 5, wherein the first, second, third, and fourth signals form part of a multipoint PSK constellation.
 7. The transmitter of claim 5, wherein the peak-to-average power ration of transmitted signals is equal to one.
 8. The transmitter of claim 5, wherein θ is selected from the group 0, π/2, π, and 3π/2. 